On the Gorenstein locus of some punctual Hilbert schemes

Let kk be an algebraically closed field and let HilbdG(PkN) be the open locus of the Hilbert scheme Hilbd(PkN) corresponding to Gorenstein subschemes. We prove that HilbdG(PkN) is irreducible for d≤9d≤9. Moreover we also give a complete picture of its singular locus in the same range d≤9d≤9. Such a description of the singularities gives some evidence to a conjecture on the nature of the singular points in HilbdG(PkN) that we state at the end of the paper.